Finite energy solutions and critical conditions of nonlinear equations in $R^n$

TL;DR

This paper investigates the critical conditions for nonlinear elliptic equations with weights, establishing invariance, existence criteria for finite energy solutions, and analyzing negative solutions in various critical regimes.

Contribution

It identifies the precise critical conditions for the existence of finite energy solutions and explores solution existence in different critical regimes for nonlinear elliptic equations.

Findings

Critical conditions correspond to invariance under scaling.

Pohozaev identity characterizes necessary and sufficient conditions.

Existence of negative solutions varies across subcritical, critical, and supercritical cases.

Abstract

This paper is concerned with the critical conditions of nonlinear elliptic equations with weights and the corresponding integral equations with Riesz potentials and Bessel potentials. We show that the equations and some energy functionals are invariant under the scaling transformation if and only if the critical conditions hold. In addition, the Pohozaev identity shows that those critical conditions are the necessary and sufficient conditions for existence of the finite energy positive solutions or weak solutions. Finally, we discuss respectively the existence of the negative solutions of the $k$-Hessian equations in the subcritical case, critical case and supercritical case. Here the Serrin exponent and the critical exponent play key roles.